Theorem 0nnq 10962

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5723	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10950	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4092	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3991	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 197	1 ⊢ ¬ ∅ ∈ Q

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2106  ∀wral 3059  ∅c0 4339   class class class wbr 5148   × cxp 5687  ‘cfv 6563  2^nd c2nd 8012  Ncnpi 10882   <_N clti 10885   ~_Q ceq 10889  Qcnq 10890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-nq 10950

This theorem is referenced by:  adderpq  10994  mulerpq  10995  addassnq  10996  mulassnq  10997  distrnq  10999  recmulnq  11002  recclnq  11004  ltanq  11009  ltmnq  11010  ltexnq  11013  nsmallnq  11015  ltbtwnnq  11016  ltrnq  11017  prlem934  11071  ltaddpr  11072  ltexprlem2  11075  ltexprlem3  11076  ltexprlem4  11077  ltexprlem6  11079  ltexprlem7  11080  prlem936  11085  reclem2pr  11086